Completeness Theorems for Logic with a Single Type Christopher Menzel (Under review at Review of Symbolic Logic) 1 Common Logic Common Logic (CL) is a standardized logical framework that arose out of the need to represent and share information on high-speed computer networks, the World Wide Web in particular. Arguably, however, its non-standard syntactic and seman- tic features have rather considerable philosophical significance (Menzel 2011) — notably, CL languages, or dialects, contain only a single lexical type the members of which all denote objects of a single semantic type in any interpretation. The Common Logic standard itself (ISO/IEC JTC 1 SC 32 20071) makes solid headway in placing CL on a solid theoretical footing, but it leaves a considerable gap: it provides the syntax and semantics of CL dialects only, leaving the choice of proof theory up to the user.2 However, because of its non-standard features — notably, a number of features characteristic of second-order languages — it is important to verify that CL dialects can be outfitted with a sound and complete proof theory. That is the central purpose of this paper. In fact, two completeness theorems will 1Authorship of an ISO/IEC document is typically attributed to an otherwise anonymous sub- committee of the overseeing “joint technical committee” — hence the moniker “JTC 1 SC 32”. The present author was an active member of SC 32 for several years and was intimately involved in the development of the CL standard, although the bulk of ISO/IEC document itself was written by Dr. Patrick Hayes. 2There are several reasons for this. First, although it contains a natural first-order fragment (which shall be the focus in this paper), full CL in fact exceeds first-order logic in expressive power and, hence, full CL-validity is not recursively axiomatizable. Second, the primary motivation for CL is to serve as a general framework for representing and exchanging information rather than for theorem proving. Relatedly, third, in the typical case, information represented or exchanged in the context of the Web will be reasoned upon by automated systems based upon logics that are computationally decidable and, hence, less powerful than full first-order logic (cf., e.g., Horrocks et al. 2005). It was therefore thought best to avoid including a proof theory even for the first-order fragment, so as to avoid the marking implementations of CL that adopt a weaker proof theory as non-conformant. be provided for CL — a “direct” theorem that follows the contours of a typical Henkin completeness theorem, albeit modified to accommodate the novel features of CL; and an “indirect” theorem that proceeds by way of a translation scheme from an arbitrary CL language into a “traditional” first-order language. Completeness then follows straightaway from the completeness of traditional first-order logic by way of a few simple theorems that demonstrate that the translation scheme “pre- serves meaning” in an appropriate sense. Because the syntax for CL is stated so generally in order to accommodate a wide variety of dialects, it would be both very awkward and theoretically superfluous to try to define a proof theory and prove a completeness theorem in comparably gen- eral terms. We therefore will work in a particular dialect that is in fact included in an appendix to the ISO standard, the Common Logic Interchange Format, or CLIF — so-called because it is a descendent of, KIF, the Knowledge Interchange Format (Genesereth 1998).3 CLIF is a particularly good choice because it is well-suited to the non-traditional syntactic features afforded by CL. CLIF will initially be outfit- ted with a more or less standard-looking logical system. However, the syntactic features of CL, as manifested in CLIF, suggest two natural, easily axiomatized ex- tensions to the semantics that will introduced in §6 below. Completeness (by the first of our two methods) will be proved for the extended semantics as well. As the motivation and significance of CL’s syntax is perhaps not obvious on its face, the following section consists entirely of an informal discussion of its distinc- tive features.4 Additionally, the final (sub)section of the paper contains some brief reflections on the connection between CL and traditional first-order logic. 2 CL’s Distinctive Features and Their Motivation CL’s most distinctive features can be summed up reasonably well as: type-freedom, variable polyadicity, and “higher-order” quantification. 3KIF is in fact still broadly used, not only because it was developed at Stanford in the 1980s by an influential group of AI researchers who promoted its use, but also simply because of the ease with which KIF statements can be constructed by means of an ASCII keyboard and hence easily distributed by standard ASCII-based communication protocols like email. 4For motivations more directly related to computational issues in artificial intelligence, database theory, and the like, see Chen et al. 1993. Entirely independent of CL, Chen and his colleagues de- veloped a logic — HiLog — whose syntax and semantics are quite similar to CL’s but whose purpose was to serve as a more flexible foundation for logic programming than traditional predicate logic. 2 Type-freedom. In general, a logic is type-free, in some respect, if it does not heed one or another traditional division into strict syntactic or semantic types, or categories. For example, the individual constants, n-function symbols, and n- place predicates of a language are typically considered to constitute distinct, mu- tually disjoint lexical categories and, hence, are considered to be different syntactic types.5 Concomitantly, the semantic values of the members of those types are tra- ditionally drawn from domains D and Fn and Rn — individuals, n-place functions, n and n-place relations, respectively. As Rn is typically taken to be a subset of }(D ) n+1 and Fn a subset of }(D ), n-place functions and might overlap with n + 1-place relations. But, at the least, n-place functions (for n > 0) and n-place relations jointly constitute semantic types that are distinct from the type of individuals. However, limited forms of type freedom, at least, are well-warranted by natural language and by common practices in knowledge representation. Many knowledge representation (KR) systems presuppose a hierarchical ontology of classes that cor- respond semantically to predicates. At the same time, classes are often considered (first-order) individuals which can themselves be subjects of predication, suggest- ing a breach in the traditional division between the semantic values of individual constants and predicates. Moreover, many KR systems have a top level class EN- TITY which is itself considered an entity. Hence, ‘ENTITY is an ENTITY’, or the like, is typically taken to be a theorem of these systems, suggesting a breach in the syntactic division between individual constants and predicates as well. Likewise, a significant measure of semantic type freedom is warranted by the ubiq- uitous phenomenon of nominalization in natural language, whereby noun phrases are generated from adjectives and verb phrases via, notably, gerundive construc- tions (e.g., wise −! being wise, runs −! running). Intuitively, a predicable expres- sion — an adjective or verb phrase — and its nominalized counterpart have the same semantic value: an individual is wise just in case being wise is among her properties; she runs just in case she engages in running. But nominalizations are singular terms. Hence, to represent the information expressed by nominalization correctly, the strict division between the semantic values of predicates and denot- ing expressions must be breached. Moreover, differences in surface grammatical form aside, since they have the same meaning there is no logical reason to repre- sent them by means of the same lexical item and, hence, the corresponding lexical division between predicates and singular terms breaks down as well.6 5Individual constants can of course be identified with 0-place function symbols. 6See, e.g., Cocchiarella 1972, Menzel 1986. These logics are not only semantically type-free in the sense of Bealer 1982 but also syntactically type-free insofar as predicates also count as terms and, hence, can serve as arguments in atomic formulas. Their occurrences in such formulas, of course, 3 However, as the strict divisions between lexical types and divisions between se- mantic types begin to break down and one type bleeds into the other, a more radical perspective emerges: the traditional divisions are best represented, not as fixed, pre-existing types but rather as contextually defined roles. This is CL’s approach: the traditional divisions between lexical/semantic types are abolished entirely: There is but a single primitive syntactic category of names and a corre- sponding semantic category of things. Vestiges of the traditional syntactic divisions survive only as contextually defined syntactic roles that any name (or, more gen- erally, any term) can play; likewise for their denotations. Thus, in terms of CLIF’s syntax, any list of names between parentheses (α β1 ::: βn) can serve as both a complex term and an atomic sentence.7 Qua term, the leftmost occurrence of the name α in the expression plays the applicative role and the thing aα it denotes plays the function role; likewise, qua sentence, that occurrence of α plays the predicative role and its denotation aα the relation role. And, in both contexts, the terms βi are all playing argument roles and their denotations the object role — in the one case as the objects to which aα is applied and, in the other, the things of which aα is predicated. Moreover, as there are no restrictions whatever on the formation of atomic sentences from names, we have, in particular, that (α α) is both a legiti- mate formula and a legitimate function term and, hence, both self-predication and self-application are expressible with ease.8 A thing plays the object role simply in virtue of being an argument to (some- thing playing the role of) a function or relation, more or less as in the semantics of standard predicate logic.